When a Term Has a Given Value

Transcript

0:00Introduction

Hello everyone! Okay, a strange little fact to start with. Take two to the power x, plus two to the power minus x, and raise the whole thing to the power six. Now look at its fourth term. Put x equals zero: the fourth term is twenty. Put x equals five: twenty again. Put x equals minus one hundred: still twenty! How can a term not care about x at all? Keep that question in mind. Today the value of a term is given to us, and we work backwards to find x. Questions of this type do come up in JEE, so let's begin.

0:41The method: the term is given, find x

So, first recall the general term. T r plus one, capital T with a small r plus one written below, is the r plus one th term. It equals n C r, with the small n at the top left of capital C, times the first term to the power n minus r, times the second term to the power r. Here n is a positive integer, and r runs zero, one, up to n; the three dots mean and so on. We always count from r equals zero. Okay? Now a quick question. Suppose a question talks about the sixth term. Which value of r do you put in? Five, yes, exactly! Not six. T r plus one is the sixth term when r plus one is six, so r is five. In general, the k th term is T k, with a small k below, and r equals k minus one. Oh no, careful: r equals six is an easy slip. Got it? Now the method itself. Step one: write that term, and simplify the powers of x. With me? Step two: set the term equal to the value the question gives you, and solve for x. Sometimes the unknown is a constant instead; same idea. Right? And step three, only when needed: if the exponent, that is the power, contains log x, take logarithms of both sides. Clear?

2:03Back to the puzzle: the powers cancel

Now, our puzzle from the start. Two to the power x, plus two to the power minus x, in round brackets, to the power six. The fourth term is T four, so r is three. T four equals six C three, times two to the power x, cubed, times two to the power minus x, cubed. Okay? Now tell me: two to the power three x, times two to the power minus three x. What is that? One, correct! And six C three is twenty. Same base, so the powers add: three x minus three x is zero, and two to the power zero is one. The small dot means times. So T four is twenty times one, which is twenty, for every real x. Mystery solved: the x simply cancels! Right? And why does it cancel here? With n equals six, there are seven terms, and T four is the middle one. In the middle term, both parts get the same power, three, so they cancel completely. With me?

3:04When log x sits in the power

Now the tricky kind. Sometimes the term contains x to the power log x. Here log, with a small ten written below, means log to the base ten. So x sits in the base, and x also sits inside the power. That is hard to solve as it stands. Okay? So we give the log a short name. Put t equals log x to the base ten. Then x equals ten to the power t. Clear? Now a question for you. If we take log to the base ten of x to the power m, what do we get? Yes, exactly! The power comes down in front: m times log x, which is m times t. So taking logarithms of both sides turns a power equation into a simple equation in t. Solve for t, and then x equals ten to the power t. Got it?

3:57Example 1: 4th term of (1 + 2^x)^3 equals the 2nd term

Alright, example one. Find x, if the fourth term of one plus two to the power x, the whole thing cubed, equals its second term. Pause the video and try it. Done? Okay, let's check. Second term, so r is one. T two equals three C one, times one squared, times two to the power x. That is three times two to the power x. Okay? Fourth term, so r is three. T four equals three C three, times two to the power x, cubed. Three C three is one, so T four is two to the power three x. With me? Now we set T four equal to T two. To simplify, I want to divide both sides by two to the power x. Can we do that? Is it safe? Yes, completely safe! Two to the power x is always positive, never zero. So, two to the power three x equals three times two to the power x. Dividing, the powers subtract: three x minus x is two x. So two to the power two x equals three; the double arrow means, which gives. Right? Now take logarithms, this time to the base two; the small two below the log is the base. Two x equals log three to the base two. So x equals half of log three to the base two. Clear?

5:21Example 2: 3rd term of (x + x^(log10 x))^5 is 10^6

Now a harder one. The third term of x plus x to the power log x to the base ten, the whole thing to the power five, is ten to the power six. Find x. Pause the video and try it. Hint: put t equals log x to the base ten. Done? Okay, let's check. Third term, so r is two. T three equals five C two, times x cubed, times x to the power log x, squared. Five C two is ten. With t equals log x, the powers add: three plus two t. So T three equals ten times x to the power three plus two t. Okay? Set it equal to ten to the power six, and cancel one ten from both sides. So, x to the power three plus two t equals ten to the power five. Right? Now x is in the base, and t, which is log x, is in the power. So take log to the base ten of both sides. What do you get? Perfect! On the left, the power comes down: three plus two t, times log x, which is t. On the right, log of ten to the power five is just five. So t times, three plus two t, equals five, that is, two t squared plus three t minus five equals zero. Just a quadratic in t! With me? Factorise it: t minus one, times two t plus five, equals zero. So t equals one, or t equals minus five over two. Got it? Now, it is tempting to say: t is negative, so reject it. Should we? No, no, wait! Keep it. Here t is log x, and a log can be negative; it only means x lies between zero and one. And x equals ten to the power t, which is always positive. So t equals one gives x equals ten, and t equals minus five over two gives x equals ten to the power minus five over two. Both answers are correct. Clear?

7:23Quick revision: write the term, set it equal, take logs

So, quick revision. One: for the k th term, write T k with r equals k minus one, and simplify the powers of x. Two: set the term equal to the given value, and solve. Three: if log x sits in the power, put t equals log x to the base ten, and take logarithms. Four: keep every valid root; t may be negative, because x equals ten to the power t is always positive. You did really well today. Now go and try the practice problems on this page. All the best!